Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum

J. Differential Equations 254 (2013) 229–255

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Journal of Differential Equations www.elsevier.com/locate/jde

Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum ✩ Shengquan Liu a,b , Haibo Yu a , Jianwen Zhang a,∗ a b

School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China Department of Mathematics, Liaoning University, Shenyang 110036, PR China

a r t i c l e

i n f o

Article history: Received 24 March 2012 Available online 29 August 2012 MSC: 76W05 76N10 35B40 35B45 Keywords: Compressible magnetohydrodynamics Vacuum Large oscillation Global weak solution Large-time behavior

a b s t r a c t In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic magnetohydrodynamic flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L 2 , and that the initial density and the gradients of initial velocity/magnetic field are bounded in L ∞ and L 2 , respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Magnetohydrodynamics (MHD) concerns the motion of a conducting fluid (plasma) in an electromagnetic field with a very wide range of applications. Because the dynamic motion of the fluid and the magnetic field interact on each other and both the hydrodynamic and electrodynamic effects in the motion are strongly coupled, the problems of MHD system are considerably complicated. The governing equations for the motion of three-dimensional compressible isentropic magnetohydrodynamic ✩ This research is partly supported by NNSFC (Grant Nos. 10801111, 10971171, 11271306), the Fundamental Research Funds for the Central Universities (Grant Nos. 2010121006, 2012121005), and the Natural Science Foundation of Fujian Province of China (Grant No. 2010J05011). Corresponding author. E-mail address: [email protected] (J. Zhang).

*

0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.08.006

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

flows, derived from fluid mechanics with appropriate modifications to account for electrical forces, have the following form (see, e.g., [2,20]):

ρt + div(ρ u ) = 0,

(1.1)

(ρ u )t + div(ρ u ⊗ u ) + ∇ P (ρ ) = (∇ × H ) × H + μu + (λ + μ)∇ div u ,

(1.2)

H t − ∇ × (u × H ) = −∇ × (ν ∇ × H ),

div H = 0,

(1.3)

where x = (x1 , x2 , x3 ) ∈ R3 is the spatial variable and t  0 is the time. The unknown functions ρ , u = (u 1 , u 2 , u 3 ) ∈ R3 , P = P (ρ ) and H = ( H 1 , H 2 , H 3 ) ∈ R3 are the density, velocity, pressure and magnetic field, respectively. The constants μ and λ are the shear and bulk viscosity coefficients of the flow and satisfy the physical restrictions:

2

μ > 0 and λ + μ  0.

(1.4)

3

The constant ν > 0 is the resistivity coefficient which is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. Here we only consider the isentropic flows in which the equation of state reads

P (ρ ) = A ρ γ

with A > 0 and γ > 1

(1.5)

where A > 0 and γ > 1 are some physical parameters. In this paper, we are interested in an initial value problem of (1.1)–(1.5) subject to the following initial conditions:

(ρ , u , H )(x, 0) = (ρ0 , u 0 , H 0 )(x) for all x ∈ R3 ,

(1.6)

and the far-field behavior:

ρ (x, t ) → ρ˜ > 0,

(u , H )(x, t ) → (0, 0) as |x| → ∞,

(1.7)

where ρ˜ > 0 is the fixed reference density. There have been numerous studies on the MHD problem by many physicists and mathematicians due to its physical importance, complexity, rich phenomena and mathematical challenges, see, for example, [2,3,9,12,13,20] and the references therein. In particular, if there is no electromagnetic effect, i.e. H ≡ 0, then (1.1)–(1.5) reduce to the compressible Navier–Stokes equations for isentropic flows, which have also been studied by many people, see, for example, [4,6,7,11,16–18] among others. For multi-dimensional compressible MHD flows, Kawashima [13] first considered the global existence of smooth solutions when the initial data are close to a non-vacuum equilibrium in H 3 -norm. In the Lions’ framework [16] (see also Feireisl et al. [5]), Hu and Wang [9] studied the global existence of weak solutions, the so-called finite energy weak solution, to the compressible MHD equations with general initial data and suitably large adiabatic exponent γ when the initial energy is merely finite. The solution obtained in [9] may have large oscillations and contain vacuum, however, it possess rather little regularity and only satisfies the equations in a very weak sense. Recently, assume that the viscosity coefficients μ and λ fulfill the following additional/non-physical conditions:

 0 < μ < ξ  2μ + λ <

3 2

√ +

21

6



μ.

(1.8)

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

231

Suen and Hoff [19] proved the global existence of weak solutions of (1.1)–(1.3) when the initial data (ρ0 , u 0 , H 0 ) satisfies

⎧  ⎨ (ρ0 − ρ˜ , u 0 , H 0 ) L 2 is sufficiently small, 0 < ρ  inf ρ0 (x)  sup ρ0 (x)  ρ¯ < ∞, ⎩

u 0 L p + H 0 L p < ∞ for some p > 6.

(1.9)

It is worth mentioning that the additional restriction (1.8) on viscosity coefficients seems unsatisfactory and non-physical, and moreover, the positive lower bound of initial density in (1.9)2 indicates there is absent of vacuum initially. It is well known that the discontinuous solutions (namely, weak solutions) are fundamental and important in both the physical and mathematical theory. Moreover, as emphasized in many papers (see, e.g. [8,15,16,21]), the possible presence of vacuum is one of the major difficulties in the study of mathematical theory of compressible fluids. So, the main purpose of this paper is to study the global existence and large-time behavior of weak solutions of (1.1)–(1.7) when the initial density may contain vacuum states. A great deal of information on the partial regularity of velocity, vorticity and magnetic field will be also obtained. Our study is mainly motivated by a recent paper due to Huang, Li and Xin [11], where the authors established the global well-posedness of classical solution with large oscillations and vacuum to the Cauchy problem of three-dimensional Navier–Stokes equations for compressible isentropic flows in a very technical and subtle way. To state the main results in a precise way, we first introduce some notations and conventions which will be used throughout the paper. Let



 f dx =

f dx

and

∂i f 

R3

∂f . ∂ xi

For k ∈ Z+ and r > 1, the standard homogeneous and inhomogeneous Sobolev spaces for scalar/vector functions are denoted by (see, e.g. [1]):







L r = L r R3 , H k = W k,2 ,

 









D k,r = u ∈ L 1loc R3 ∇ k u  L r < ∞ ,

W k,r = L r ∩ D k,r ,

 D 1 = u ∈ L 6 ∇ u L 2 < ∞ ,

D k = D k,2 ,

 

u D k,r = ∇ k u  Lr .

Weak solutions of (1.1)–(1.7) are defined in a usual way. Definition 1.1. A pair of functions (ρ , u , H ) is said to be a weak solution of (1.1)–(1.7) provided that (ρ − ρ˜ , ρ u , H ) ∈ C ([0, ∞); H −1 (R3 )), u ∈ L 2loc (0, ∞; D 1 ), H ∈ L ∞ (0, ∞; L 2 ) ∩ L 2 (0, ∞; D 1 ), and div H (·, t ) = 0 in D (R3 ) for t > 0. Moreover, the following identities hold for any test function ψ ∈ D(R3 × (t 1 , t 2 )) with t 2  t 1  0 and j = 1, 2, 3:



t2 t2

(ρ ψt + ρ u · ∇ψ) dx dt , ρ ψ(x, t ) dx = t1

t1



t2

ρ u j ψ(x, t ) dx

+

t2





μ∇ u j · ∇ψ + (μ + λ)(div u )ψx j dx dt

t1

t1

t2



1



ρ u j ψt + ρ u j u · ∇ψ + P (ρ )ψx j + | H |2 ψx j − H j H · ∇ψ dx dt ,

=

2

t1

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

and



t2 t2



H ψ(x, t ) dx = H j ψt + H j u · ∇ψ − u j H · ∇ψ − ν ∇ H j · ∇ψ dx dt . j

t1

t1

For any given initial data (ρ0 , u 0 , H 0 ), we define the initial energy C 0 as follows:

  C0 

1 2



1

ρ0 |u 0 |2 + | H 0 |2 + G (ρ0 ) dx, 2

(1.10)

where G (ρ ) is the potential energy density defined by

ρ G (ρ )  ρ ρ˜

P (s) − P (ρ˜ ) s2

ds.

(1.11)

It is clear that

c 1 (ρ˜ , ρ¯ )(ρ − ρ˜ )2  G (ρ )  c 2 (ρ˜ , ρ¯ )(ρ − ρ˜ )2

for ρ˜ > 0, 0  ρ  2ρ¯ ,

where c 1 , c 2 are positive constants depending only on ρ˜ and ρ¯ . We are now ready to state our main results. Theorem 1.1. For given positive numbers M 1 , M 2 (not necessarily small) and ρ¯  ρ˜ + 1, assume that the initial data (ρ0 , u 0 , H 0 ) satisfies



0  inf ρ0  sup ρ0  ρ¯ , div H 0 = 0,

∇ u 0 2L 2  M 1 , ∇ H 0 2L 2  M 2 .

(1.12)

Then there exists a positive constant ε , depending on μ, λ, ν , γ , A, ρ˜ , ρ¯ , M 1 and M 2 , such that if

C 0  ε,

(1.13)

then there exists a weak solution (ρ , u , H ) of (1.1)–(1.7) in the sense of Definition 1.1 satisfying

0  ρ (x, t )  2ρ¯

for all x ∈ R3 , t  0,

(1.14)

and

 lim

t →∞

 |ρ − ρ˜ | p + ρ 1/2 |u |4 + | H |q dx = 0,



(1.15)

where p ∈ (2, +∞) and q ∈ (2, 6]. Remark 1.1. Compared with the results obtained in [19], the initial vacuum now is allowed. Moreover, it is worth mentioning that D 1 only embeds into L 6 , but not into L p with p > 6.

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

233

Theorem 1.1 will be proved by constructing weak solutions as limits of smooth solutions. Roughly speaking, we first utilize Kawashima’s theorem (see Lemma 2.4) to guarantee the local existence of smooth solutions with strictly positive density, then extend the smooth non-vacuum solutions globally in time just under the condition that the initial energy is suitably small (see Theorem 4.1), and finally let the lower bound of the initial density go to zero. So, for the proof of Theorem 1.1, it suffices to derive some global a priori estimates which are independent of the lower bound of density. However, due to the presence of vacuum states, the analysis (especially, the energy estimates of A 1 ( T ), A 2 ( T )) here is completely different than that in [19]. To overcome the difficulties induced by vacuum, we shall make use of some ideas developed in [11]. As that in [11], it turns out that the key step is to obtain the time-independent upper bound of the density. However, because of the influence of magnetic field and its interaction with the hydrodynamic motion, the problem of MHD considered becomes more complicated than that of Navier–Stokes equations. For example, since the material derivative u˙  ut + u · ∇ u is strongly associated with the density in the presence of vacuum, some additional difficulties will arise when we deal with the magnetic force (∇ × H ) × H and the convection term ∇ × (u × H ). These difficulties will be circumvented by using Sobolev inequalities and the important relations among the velocity u, pressure P , magnetic field H , vorticity ω and effective viscous flux F (see Lemma 2.2) in a subtle way. The remainder of this paper is organized as follows. We first collect some useful inequalities and basic results in Section 2. The global-in-time a priori estimates will be proved in Section 3. Finally, Theorem 1.1 will be proved in Section 4 by constructing weak solutions as limits of smooth solutions. 2. Auxiliary lemmas In this section, we state some auxiliary lemmas, which will be frequently used in the sequel. We start with the well-known Gagliardo–Nirenberg inequality (see, for instance, [1,14]). Lemma 2.1. Assume that f ∈ H 1 and g ∈ L q ∩ D 1,r with q > 1 and r > 3. Then for any p ∈ [2, 6], there exists a positive constant C , depending only on p, q and r, such that (6− p )/(2p )

f L p  C f L2

(3p −6)/2p

∇ f L 2

q(r −3)/(3r +q(r −3))

g L ∞  C g Lq

3r /(3r +q(r −3))

∇ g Lr

(2.1)

, .

(2.2)

As it was pointed out in [4–7,16], the effective viscous flux plays an important role in the mathematical theory of compressible fluid dynamics. However, due to the additional presence of magnetic field, we need to define the effective viscous flux in a slightly different manner. More precisely, let F and ω be the (modified) effective flux and vorticity defined by





F  (2μ + λ) div u − P (ρ ) − P (ρ˜ ) −

1 2

| H |2 and ω  ∇ × u .

(2.3)

Due to div H = 0, one has

1

(∇ × H ) × H = H · ∇ H − ∇| H |2 . 2

So, it follows from (1.2) that

 F = div(ρ u˙ − H · ∇ H ),

μω = ∇ × (ρ u˙ − H · ∇ H ),

(2.4)

where “ ˙ ” denotes the material derivative, i.e.,

˙f := ∂t f + u · ∇ f .

(2.5)

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

In view of Lemma 2.1 and the classical estimates of elliptic system, we have Lemma 2.2. There exists a generic positive constant C , depending only on μ and λ, such that for any 2  p  6,



∇ F L p + ∇ ω L p  C ρ u˙ L p + H · ∇ H L p ,  



∇ u L p  C F L p + ω L p +  P (ρ ) − P (ρ˜ ) L p + H 2L 2p .

(2.6) (2.7)

Proof. An application of the L p -estimate of elliptic systems to (2.4) gives (2.6). On the other hand, since −u = −∇ div u + ∇ × ω , it holds that

∇ u = −∇(−)−1 ∇ div u + ∇(−)−1 ∇ × ω, which, combined with the standard L p -estimate and (2.3), yields that ∀ p ∈ [2, 6],



∇ u L p  C div u L p + ω L p  

  C F L p + ω L p +  P (ρ ) − P (ρ˜ ) L p + H 2L 2p . This finishes the proof of Lemma 2.2.

2

The next lemma is due to Zlotnik [22], which will be used to prove the uniform (in time) upper bound of density. Lemma 2.3. Assume that y ∈ W 1,1 (0, T ) solves the ODE system:

y = g ( y ) + b (t ) on [0, T ],

y (0) = y 0 ,

(2.8)

where b ∈ W 1,1 (0, T ), g ∈ C (R) and g (∞) = −∞. If there are two non-negative numbers N 0 , N 1  0 satisfying

b(t 2 ) − b(t 1 )  N 0 + N 1 (t 2 − t 1 )

for all 0  t 1 < t 2  T ,

(2.9)

then it holds that





y (t )  max y 0 , ξ ∗ + N 0 < ∞

on [0, T ],

(2.10)

where ξ ∗ ∈ R is a constant such that

g (ξ )  − N 1

for ξ  ξ ∗ .

(2.11)

Finally, we need the local-in-time existence theorem of (1.1)–(1.7) in the case that the initial density is strictly away from vacuum (see [13]). Lemma 2.4. Assume that the initial data (ρ0 , u 0 , H 0 ) satisfies

(ρ0 − ρ˜ , u 0 , H 0 ) ∈ H 3 ,

div H 0 = 0

and

inf ρ0 > 0.

(2.12)

Then there exists a positive time T 0 , which may depend on inf ρ0 , such that the Cauchy problem (1.1)–(1.7) has a unique smooth solution (ρ , u , H ) on R3 × [0, T 0 ] satisfying

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

235

ρ (x, t ) > 0 for all x ∈ R3 , t ∈ [0, T 0 ],

ρ − ρ˜ ∈ C [0, T 0 ]; H

 3

(2.13)

1

 ∩ C [0, T 0 ]; H 2 ,

(2.14)

and





 (u , H ) ∈ C [0, T 0 ]; H 3 ∩ C 1 [0, T 0 ]; H 1 ∩ L 2 [0, T 0 ]; H 4 .

(2.15)

3. A priori estimates This section is devoted to the global-in-time a priori estimates. To begin, let T > 0 be fixed and assume that (ρ , u , H ) is a smooth solution of (1.1)–(1.7) defined on R3 × (0, T ]. For simplicity, we set σ (t )  min{1, t } and define

A 1 ( T )  sup

0t  T



T



σ ∇ u 2L 2 + ∇ H 2L 2 +







 

2 2 σ ρ 1/2 u˙ L 2 + H t 2L 2 + ∇ 2 H L 2 dt ,

(3.1)

0

2 2 

  A 2 ( T )  sup σ ρ 1/2 u˙  L 2 + ∇ 2 H  L 2 + H t 2L 2 +

T

0t  T





σ 2 ∇ u˙ 2L 2 + ∇ H t 2L 2 dt

2

(3.2)

0

and



A 3 ( T )  sup



∇ u 2L 2 + ∇ H 2L 2 .

0t  T

(3.3)

The proof of Theorem 1.1 is based on the following energy estimates of (u , H ) and uniform upper bound of ρ . Proposition 3.1. Assume that (ρ0 , u 0 , H 0 ) satisfies (1.12) and that (ρ , u , H ) is a smooth solution of (1.1)– (1.7) on R3 × (0, T ]. There exist two positive constants ε and K , depending on μ, λ, ν , γ , A, ρ¯ , ρ˜ , M 1 and M 2 , such that if



0  ρ (x, t )  2ρ¯

for (x, t ) ∈ R3 × [0, T ], 1/2

A 1 ( T ) + A 2 ( T )  2C 0 ,

A3





σ ( T )  3K ,

(3.4)

then one has



0  ρ (x, t )  7ρ¯ /4 A1 (T ) + A2 (T ) 

for (x, t ) ∈ R3 × [0, T ],

1/2 C0 ,

A3





σ ( T )  2K ,

(3.5)

provided that

C 0  ε. Proof. As a result of Lemmas 3.2, 3.5 and 3.7, one gets (3.5) provided K and in Lemmas 3.2 and 3.7, respectively. 2

(3.6)

ε are chosen as the ones

For simplicity, throughout this section we denote by C the various generic positive constants, which may depend on μ, λ, ν , γ , A, ρ¯ , ρ˜ , M 1 and M 2 , but are independent of T . We also sometimes write C (α ) to emphasize the dependence on α . We begin the proof of Proposition 3.1 with the standard energy estimate of (ρ , u , H ).

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

Lemma 3.1. Let (ρ , u , H ) be a smooth solution of (1.1)–(1.7) on R3 × (0, T ]. Then,

  G (ρ ) +

sup

0t  T

T +



1 2



1

ρ |u |2 + | H |2 dx 2



μ ∇ u 2L 2 + (λ + μ) div u 2L 2 + ν ∇ H 2L 2 dt  C 0 .

(3.7)

0

Proof. Due to div H = 0, it is easy to check that

∇ × (u × H ) = H · ∇ u − u · ∇ H − H div u , 1

(∇ × H ) × H = H · ∇ H − ∇| H |2 and −∇ × (∇ × H ) =  H . 2

Thus, multiplying (1.1), (1.2) and (1.3) by G (ρ ), u and H , respectively, integrating the resulting equations by parts over R3 × (0, T ), and adding them together, one easily obtains (3.7). 2 The next lemma is concerned with the estimate of A 3 ( T ), which plays an important role in the proofs of A 1 ( T ), A 2 ( T ) and the uniform upper bound of density. Lemma 3.2. Suppose that the conditions of Proposition 3.1 hold. Then there exist positive constants K and ε1 , depending on μ, λ, ν , γ , A, ρ¯ , ρ˜ , M 1 and M 2 , such that



 A3 σ (T ) +

σ( T )

  

 1/2 2 ρ u˙  2 + H t 22 + ∇ 2 H 22 dt  2K , L L L

(3.8)

0

and moreover,

T sup

0t  T

∇ H 2L 2

+

2  

H t 2L 2 + ∇ 2 H L 2 dt  C ∇ H 0 2L 2 ,



(3.9)

0

provided A 3 (σ ( T ))  3K and C 0  ε1 . Proof. Multiplying (1.2) by ut and integrating by parts over R3 , we obtain

1 d

2 dt

=





2

μ ∇ u 2L 2 + (μ + λ) div u 2L 2 + ρ 1/2 u˙ L 2  

d dt



+

1 2





 | H |2 (div u ) − H · ∇ u · H + P (ρ ) − P (ρ˜ ) (div u ) dx

 −



H t · ∇ u · H + H · ∇ u · H t − H · H t (div u ) dx

 P t (div u ) dx +

ρ u · ∇ u · u˙ dx.

On the other hand, it follows from (1.3) that

(3.10)

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

ν

237

 2 

∇ H 2L 2 + H t 2L 2 + ν 2 ∇ 2 H L 2 dt   2 = | H t − ν  H | dx = | H · ∇ u − u · ∇ H − H div u |2 dx. d

(3.11)

Thus, adding (3.10) and (3.11) together gives



d dt

1 2

∇ u 2L 2

μ



1

+ (μ 2

+ λ) div u 2L 2

+ν

∇ H 2L 2

2  2 

 + ρ 1/2 u˙ L 2 + H t 2L 2 + ν 2 ∇ 2 H L 2   

 d 1 = | H |2 (div u ) − H · ∇ u · H + P (ρ ) − P (ρ˜ ) (div u ) dx dt



+

2



 + 

d dt





H t · ∇ u · H + H · ∇ u · H t − H · H t (div u ) dx −

P t (div u ) dx



ρ u · ∇ u · u˙ dx + I0 +

4 

| H · ∇ u − u · ∇ H − H div u |2 dx

Ii.

(3.12)

i =1

By Lemma 2.1 and Cauchy–Schwarz inequality, we have for any

η > 0 that

I 1  C H L ∞ H t L 2 ∇ u L 2

1/2 1/2   C ∇ H L 2 ∇ 2 H  L 2 H t L 2 ∇ u L 2 2

   η ∇ 2 H L 2 + H t 2L 2 + C (η) ∇ u 4L 2 ∇ H 2L 2 .

(3.13)

To deal with I 2 , we first observe from (1.1) and (2.3) that





P (ρ ) = A ρ γ

P t = −u · ∇ P − P (ρ˜ ) − γ P div u ,

(3.14)

and

div u =



1 2μ + λ





F + P − P (ρ˜ ) +

1 2

 | H |2 .

So, after integrating by parts we infer from (2.6), (3.4) and (3.7) that

 

γ P (div u )2 +

I2 =

1







F + P − P (ρ˜ ) +

1

 

 | H |2 u · ∇ P − P (ρ˜ ) dx

2μ + λ 2    

 4  C ∇ u 2L 2 +  P − P (ρ˜ ) L 4 + C ∇ u L 2 ∇ F L 2 + H ∇ H L 2  P − P (ρ˜ ) L 3 



  1/3  C ∇ u 2L 2 + C 0 + C C 0 ∇ u L 2 ρ 1/2 u˙  L 2 + H ∇ H L 2 2 2  

 

4/3  η ρ 1/2 u˙  L 2 + ∇ 2 H L 2 + C (η) C 0 + ∇ u 2L 2 + ∇ u L 2 ∇ H 2L 2 ,



(3.15)

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

where we have also used the following inequality (due to (2.1)):

1/2 3/2 

H 2L 12  C H ∇ H L 2  C H L 6 ∇ H L 3  C ∇ H L 2 ∇ 2 H L 2 .

(3.16)

Using (2.1), (2.6), (2.7), (3.4), (3.7) and (3.16), we deduce that









I 3  ρ 1/2 u˙  L 2 ∇ u L 3 u L 6  ρ 1/2 u˙  L 2 ∇ u L 2 ∇ u L 6 3/2

1/2

 1/2   1/2  3/2  1/2  ρ 1/2 u˙ L 2 ∇ u L 2 ρ 1/2 u˙  L 2 + H ∇ H L 2 +  P − P (ρ˜ ) L 6  1/2 1/4    3/2

3/4   ρ 1/2 u˙ L 2 ∇ u L 2 1 + ρ 1/2 u˙ L 2 + ∇ H L 2 ∇ 2 H L 2 2 2  

 

 η ρ 1/2 u˙  2 + ∇ 2 H  2 + C (η) ∇ u 32 + ∇ u 62 + ∇ u 42 ∇ H 22 , L

L

L

L

L

L

(3.17)

and finally,



I 4  C u 2L 6 ∇ H 2L 3 + ∇ u 2L 2 H 2L ∞



   C ∇ u 2L 2 ∇ H L 2 ∇ 2 H L 2  2  η∇ 2 H  L 2 + C (η) ∇ u 4L 2 ∇ H 2L 2 .

(3.18)

Thanks to (3.4), (3.7) and (2.1), we find









1/2 1/2 3/2 I 0  C ∇ u L 2  P − P (ρ˜ ) L 2 + H 2L 4  C ∇ u L 2 C 0 + H L 2 ∇ H L 2



1/2 

1/4 3/2  1/2  C ∇ u L 2 C 0 + C 0 ∇ H L 2  η ∇ u 2L 2 + C (η) C 0 + C 0 ∇ H 3L 2 .

(3.19)

Thus, putting (3.13), (3.15), (3.17) and (3.18) into (3.12), integrating it over (0, σ ( T )), and using Cauchy–Schwarz inequality, by virtue of (3.4), (3.7) and (3.19) we conclude that (choosing η > 0 sufficiently small)



 A3 σ (T ) +

σ( T )

  

 1/2 2 ρ u˙  2 + H t 22 + ∇ 2 H 22 dt L L L

0



 1/2  C 1 + ∇ u 0 2L 2 + ∇ H 0 2L 2 + C C 0

sup

0t σ ( T )

∇ H 3L 2

σ( T )



∇ u 4L 2 ∇ u 2L 2 + ∇ H 2L 2 dt

+C 0

σ( T )

 C (M1 , M2 ) +

1/2 3/2

C C0 A3

 σ (T ) + C

 C (M1 , M2 ) +

1/2 3/2

C C0 A3

  σ ( T ) + C C 0 A 23 σ ( T )

 1/2  K + C C 0 A 23 σ ( T ) ,

sup

0t σ ( T )



∇ u 4L 2



∇ u 2L 2 + ∇ H 2L 2 dt

0

(3.20)

where K  C ( M 1 , M 2 ). As an immediate result of (3.20), one obtains (3.8) provided it holds that A 3 (σ ( T ))  3K and C 0  ε1,1  min{1, (9C K )−2 }.

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

239

To prove (3.9), we deduce from (3.11) and (3.18) that

T sup

0t  T

∇ H 2L 2

+

 2 

H t 2L 2 + ∇ 2 H L 2 dt



0

T 

C ∇ H 0 2L 2

+ C 1 sup

0t  T

∇ H 2L 2

∇ u 4L 2 dt .

(3.21)

0

On the other hand, it follows from (3.4), (3.7) and (3.8) that

T

σ( T )

∇ u 4L 2

dt =

0

T

∇ u 4L 2

∇ u 4L 2 dt

dt + σ (T )

0

σ( T )



sup

0t σ ( T )

∇ u 2L 2

∇ u 2L 2

dt +

0

sup



σ ( T )t σ ( T )

σ

∇ u 2L 2

 C ( K )C 0 .



T

∇ u 2L 2 dt

σ (T )

(3.22)

Thus, if C 0 is chosen to be such that



C 0  ε1  min

 −1 



ε1,1 , 2C 1 C ( K )

,

then substituting (3.22) into (3.21) immediately leads to (3.9). The proof of Lemma 3.2 is therefore complete. 2 Next we derive preliminary bounds for A 1 ( T ) and A 2 ( T ). Lemma 3.3. Assume that the conditions of Proposition 3.1 hold. Then,

T A1 (T )  C C 0 +

4



σ 2  P − P (ρ˜ )L 4 dt

(3.23)

0

and

T

σ 2 ∇ u 4L 4 dt ,

A2 (T )  C C 0 + C A1 (T ) + C

(3.24)

0

provided C 0  ε1 . Proof. Multiplying (3.12) by σ (t ), integrating the resulting equation over (0, T ), and using the similar arguments as those in the derivations of (3.13), (3.15) and (3.17)–(3.19), we infer from (3.4), (3.7), (3.9), (3.22) and Cauchy–Schwarz inequality that

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

T A1 (T )  C C 0 + C





∇ u 2L 2 + ∇ H 2L 2 + ∇ u 4L 2 dt

0

+

1/2 C C0

T +C



sup ∇ H L 2 sup

0t  T

0t  T



σ ∇ H 2L 2





4 

σ ∇ u 6L 2 + σ ∇ u 4L 2 ∇ H 2L 2 + σ 2  P − P (ρ˜ )L 4 dt

0

1

T







σ

| H |2 (div u ) − H · ∇ u · H + P (ρ ) − P (ρ˜ ) (div u )

dx dt

+C

2

0

1/2 C C0

 C C0 + T +C

sup

0t  T

σ



∇ u 2L 2

+ ∇ H 2L 2



T +C

4



σ 2  P − P (ρ˜ )L 4 dt

0





2 

σ ∇ u 2L 2 + H 4L 4 +  P − P (ρ˜ )L 2 dt

0

T



4

σ 2  P − P (ρ˜ )L 4 dt ,

 C C0 + C 0

which proves (3.23). Note here that 0  σ  1 for t  0 and σ = 0 for t > 1. To estimate A 2 ( T ), applying the operator σ 2 u˙ j [∂t + div(u ·)] to both sides of j-th equation of (1.2) and integrating them by parts over R3 , we obtain after summing up that

L

1 d





σ 2 ρ |u˙ |2 dx − μ

2 dt









 2







σ 2 u˙ j ∂ j P t + div(∂ j P u ) dx 





σ 2 u˙ j ∂ j H i H ti + div H i ∂ j H i u dx

−  +





σ σ ρ |u˙ | dx −

=



σ 2 u˙ j ∂t ∂ j div u + div(u ∂ j div u ) dx

− (λ + μ) 



σ 2 u˙ j utj + div u u j dx









σ 2 u˙ j ∂t H i ∂i H j + div H i ∂i H j u dx 

4 

Ri.

(3.25)

i =1

Here and in what follows, we use the Einstein convention that repeated indices denote the summation over the indices. We are now in a position of estimating some terms in (3.25). First, the second term on the lefthand side can be estimated from below as follows:

 −μ







σ 2 u˙ j utj + div u u j dx

S. Liu et al. / J. Differential Equations 254 (2013) 229–255







σ 2 ∂k u˙ j ∂k utj − ∂ik2 u˙ j u i ∂k u j − ∂i u˙ j ∂k u i ∂k u j dx

=μ 





σ 2 |∇ u˙ |2 + ∂k u˙ j ∂i u i ∂k u j − ∂k u˙ j ∂k u i ∂i u j − ∂i u˙ j ∂k u i ∂k u j dx

=μ 7μ



241

8

σ 2 ∇ u˙ 2L 2 − C σ 2 ∇ u 4L 4 .

In a similar manner,







σ 2 u˙ j ∂t ∂ j div u + div(u ∂ j div u ) dx 

−(λ + μ)

μ+λ 2

σ 2 div u˙ 2L 2 − C σ 2 ∇ u 4L 4 ,

and consequently,

1 d

L



σ 2 ρ |u˙ |2 dx +

2 dt

7μ 8

σ 2 ∇ u˙ 2L 2 − C σ 2 ∇ u 4L 4 .

(3.26)

For the second term R 2 on the right-hand side, using (3.14) and integrating by parts, by (3.4) and Cauchy–Schwarz inequality we have













σ 2 −γ P (ρ )∂i u i ∂ j u˙ j + P (ρ )∂i u i ∂ j u˙ j − P (ρ )∂ j u i ∂i u˙ j dx

R2 =







σ 2 −γ P (ρ )∂i u i ∂ j u˙ j + P (ρ )∂i u i ∂ j u˙ j − P (ρ )∂ j u i ∂i u˙ j dx

=

μ



8

σ 2 ∇ u˙ 2L 2 + C σ 2 ∇ u 2L 2 .

(3.27)

By virtue of (2.1), (3.7) and (3.9), we obtain after integrating by parts that



R 3  C σ 2 ∇ u˙ L 2 H L 3 H t L 6 + u L 6 H L 6 ∇ H L 6



  

1/2 1/2  C σ 2 ∇ u˙ L 2 H L 2 ∇ H L 2 ∇ H t L 2 + ∇ u L 2 ∇ H L 2 ∇ 2 H L 2  2 

1/2 μ  σ 2 ∇ u˙ 2L 2 + C σ 2 C 0 ∇ H t 2L 2 + ∇ u 2L 2 ∇ 2 H L 2 , 8

(3.28)

and similarly, using the fact that div H = 0, we have

 R4 = −



μ 8









σ 2 ∂i u˙ j H j H ti + H i H tj + ∂k u˙ j H i ∂i H j uk dx



2 

σ 2 ∇ u˙ 2L 2 + C σ 2 C 01/2 ∇ H t 2L 2 + ∇ u 2L 2 ∇ 2 H L 2 .

(3.29)

Thus, putting (3.26)–(3.29) into (3.25) and integrating the resulting inequality over (0, T ), we infer from (3.4) and (3.7) that

sup

0t  T





2 

σ 2 ρ 1/2 u˙ L 2 +

T

σ 2 ∇ u˙ 2L 2 dt 0

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

T C

 1/2 2  σ ρ u˙ L 2 + σ 2 ∇ u 2L 2 dt + C

T

0

σ 2 ∇ u 4L 4 dt 0

1/2

T



σ 2 ∇ H t 2L 2 dt + C sup σ ∇ u 2L 2

+ C C0



0t  T

0

T



2

σ ∇ 2 H L 2 dt

0

T

σ 2 ∇ u 4L 4 dt .

 C C 0 + C A1 (T ) + C

(3.30)

0

Differentiating (1.3) with respect to t and multiplying the resulting equations by obtain after integrating by parts over R3 × (0, T ) that

1



sup

2 0t  T

T



σ 2 H t 2L 2 + ν

σ 2 ∇ H t 2L 2 dt 0

T =

σ 2 H t in L 2 , we

σσ



T

H t 2L 2

σ 2 ( H · ∇ u˙ · H t + u˙ · ∇ H t · H ) dx dt

dt +

0

0

T +







σ 2 H i ∂i H tj − H k ∂ j H tk u · ∇ u j dx dt

0

T

σ 2 ( H t · ∇ u − H t div u − u · ∇ H t ) · H t dx dt 

+

4 

Ri,

(3.31)

i =1

0

where we have used div H = 0, ut = u˙ − u · ∇ u and



 u˙ · ∇ H · H t dx =

−





H · H t (div u˙ ) + u˙ · ∇ H t · H dx.

The second term on the right-hand side of (3.31) can be estimated as follows, using (2.1), (3.4), (3.7) and (3.9):

T R2  C





σ 2 H 1L 2/2 ∇ H 1L 2/2 H t L 6 ∇ u˙ L 2 + u˙ L 6 ∇ H t L 2 dt

0



ν

T

σ

4

2

∇ H t 2L 2

dt +

1/2 C C0

0

 C C0 +

T

σ 2 ∇ u˙ 2L 2 dt 0

ν

T

σ 2 ∇ H t 2L 2 dt .

4 0

In view of (3.4), (3.9), (3.16) and (3.22), we have

(3.32)

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

243

T 3

σ 2 H L 12 ∇ H t L 2 u L 6 ∇ u L 4 dt

R C 0

T C



1/4

σ 2 ∇ 2 H L 2 ∇ H t L 2 ∇ u L 2 ∇ u L 4 dt

0



ν

T

T

σ

4

2

∇ H t 2L 2

dt + C

0

σ

2

∇ u 4L 4

 

 dt + C sup σ ∇ 2 H  L 2 0t  T

0

ν

 C C0 +

T

∇ u 4L 2 dt 0

T

σ 2 ∇ H t 2L 2 dt + C

4

T

0

σ 2 ∇ u 4L 4 dt ,

(3.33)

0

and similarly,

T 4





σ 2 H t 2L 4 ∇ u L 2 + u L 6 ∇ H t L 2 H t L 3 dt

R C 0

T C

σ 2 H t 1L 2/2 ∇ H t 3L 2/2 ∇ u L 2 dt

0



ν

T

σ

4

2

∇ H t 2L 2

dt + C sup

0

 C C0 +



0t  T

σ

2

∇ u 2L 2

T

H t 2L 2 dt 0

T

ν

σ 2 ∇ H t 2L 2 dt .

4

(3.34)

0

Thus, combining (3.32)–(3.34) with (3.31) shows that

sup

0t  T



σ

2

H t 2L 2



T +

T

σ

2

∇ H t 2L 2

0

σ 2 ∇ u 4L 4 dt .

dt  C C 0 + C A 1 ( T ) + 0

Moreover, it follows from (1.3), the L 2 -estimate of elliptic system and Lemma 2.1 that

 2 2

 ∇ H  2  C H t 22 + u ∇ H 22 + H ∇ u 22 L L L L

  C H t 2L 2 + u 2L 6 ∇ H 2L 3 + H 2L ∞ ∇ u 2L 2   

 C H t 2L 2 + ∇ u 2L 2 ∇ H L 2 ∇ 2 H  L 2 2

 1  ∇ 2 H L 2 + C H t 2L 2 + ∇ u 4L 2 ∇ H 2L 2 , 2

which, together with (3.30) and (3.35), gives

(3.35)

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255



A 2 ( T )  C C 0 + C A 1 ( T ) + C sup

0t  T

σ

2 

∇ u 2L 2 ∇ H 2L 2 +

T

σ 2 ∇ u 4L 4 dt

C 0

T

σ 2 ∇ u 4L 4 dt ,

 C C 0 + C A1 (T ) + C 0

where we have also used (3.4) and (3.9). The proof of Lemma 3.3 is therefore complete.

2

Clearly, we still need to deal with ∇ u L 4 and P − P (ρ˜ ) L 4 . Lemma 3.4. Let F , ω be the ones defined in (2.3), and let the conditions of Proposition 3.1 be satisfied. Then,

T





4 

σ 2 ∇ u 4L 4 + F 4L 4 + ω 4L 4 +  P − P (ρ˜ )L 4 dt  C C 03/4 ,

(3.36)

0

provided C 0  ε1 . Proof. In view of the standard L p -estimate, we have

T

T

σ

2

∇ u 4L 4

0









σ 2 div u 4L 4 + ω 4L 4 dt

dt  C 0

T C

4



σ 2 F 4L 4 + ω 4L 4 +  P − P (ρ˜ )L 4 + H 8L 8 .

(3.37)

0

The right-hand side of (3.37) will be estimated term by term as follows. First, it follows from (2.3), (3.4), (3.7), (3.8) and (3.9) that

  

F L 2 + ω L 2  ∇ u L 2 +  P − P (ρ˜ ) L 2 + H 2L 4  C ,

(3.38)

and hence, using Lemmas 2.1, 2.2, (3.4), (3.7), (3.9), (3.16) and (3.38), we see that

T

σ

2



F 4L 4

+ ω

4L 4



T

0







σ 2 F L 2 + ω L 2 ∇ F 3L 2 + ∇ ω 3L 2 dt

dt  C 0

T C

σ



2  1/2

ρ

3 u˙  2 dt + C L

0

 

  C sup σ ρ 1/2 u˙ L 2 0t  T

T 0

T

σ ρ 1/2 u˙ 2L 2 dt 0



3/2

σ 2 ∇ H 9L 2/2 ∇ 2 H L 2 dt

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

 3/2

 + C sup σ ∇ 2 H L 2

T

0t  T

∇ H 2L 2 dt 0

1/2 C A2 (T ) A1 (T ) +



245

3/4

C C0  C C0 .

(3.39)

Next, multiplying (3.14) by 3σ 2 ( P − P (ρ˜ ))2 , integrating the resulting equation by parts over R3 × (0, T ) and using the effective viscous flux F , we obtain

T

1 2μ + λ



4

σ 2  P − P (ρ˜ )L 4 dt

0





T

3

σ 2 P − P (ρ˜ ) (·, T ) dx − 2

=



σσ



3

P − P (ρ˜ )

dx dt

0

−

T

1 2μ + λ

T



σ2



3

P − P (ρ˜ )

F+

1 2

 | H |2 dx dt

0





2

γ P (div u ) P − P (ρ˜ ) dx dt

σ2

+3



0

T  C C0 + C



3



σ 2  P − P (ρ˜ )L 4 F L 4 + H 2L 8 dt

0

T +C

2



σ 2  P − P (ρ˜ )L 4 ∇ u L 2 dt

0



3/4 C (δ)C 0

T +δ

σ



2

4 P − P (ρ˜ ) L 4 dt + C (δ)

0

T

σ 2 H 8L 8 dt , 0

where we have used (3.4), (3.7), (3.39) and Cauchy–Schwarz inequality. Noting that,

T

T

σ

2

H 8L 8

σ 2 H 4L ∞ H 4L 4 dt

dt  C

0

0

T C



2

σ 2 H L 2 ∇ H 5L 2 ∇ 2 H L 2 dt

0

1/2

T

 C0

0



2

σ 2 ∇ 2 H L 2 dt  C C 0

(3.40)

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

due to Lemma 2.1, (3.4), (3.7) and (3.9). This, together with (3.40), gives (choosing δ > 0 small enough)

T







4 σ 2  P − P (ρ˜ )L 4 + H 8L 8 dt  C C 03/4 .

(3.41)

0

Thus, combining (3.37) with (3.39) and (3.41) leads to the desired estimate of (3.36).

2

Lemma 3.5. Suppose that the conditions of Proposition 3.1 hold. Then there exists a positive constant depending on μ, λ, ν , γ , A, ρ¯ , ρ˜ , M 1 and M 2 , such that 1/2

A1 (T ) + A2 (T )  C 0 ,

ε2 ,

(3.42)

provided C 0  ε2 . Proof. Indeed, it follows from (3.23), (3.24) and (3.36) that 3/4

A1 (T ) + A2 (T )  C C 0 . Thus, if C 0 is chosen to be such that



C 0  ε2  min then one immediately obtains (3.42).



ε1 , C −4 ,

2

In order to complete the proof of Proposition 3.1, we still need to estimate the upper bound of density. To this end, we first prove Lemma 3.6. Suppose that the conditions of Proposition 3.1 hold. Then,

sup

0t  T





∇ u 2L 2 + ∇ H 2L 2 +

T

  

 1/2 2 ρ u˙  2 + H t 22 + ∇ 2 H 22 dt  C , L L L

(3.43)

0

2 2  

 sup σ ρ 1/2 u˙  L 2 + H t 2L 2 + ∇ 2 H  L 2 +

0t  T

T





∇ u˙ 2L 2 + ∇ H t 2L 2 dt  C ,

(3.44)

0

provided C 0  ε2 . Proof. As an immediate consequence of Lemmas 3.2 and 3.5, one gets (3.43). The proof of (3.44) is similar to the one used for (3.24). More precisely, applying the operator σ u˙ [∂t + div(u ·)] to both sides of (1.2) and integrating the resulting equation by parts over R3 × (0, T ), we deduce from (3.7), (3.42), (3.43) as well as (3.26) and (3.27) (with σ 2 replaced by σ ) that

sup

0t  T

 1/2 2  σ ρ u˙  2 +

T

σ ∇ u˙ 2L 2 dt

L

0

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

T C +



247



 |∇ u |4 + |∇ u˙ || H || H t | + |∇ u˙ || H ||u ||∇ H | dt

σ 0

T





σ ∇ u 4L 4 + ∇ u˙ L 2 H L ∞ H t L 2 + ∇ u˙ L 2 H L 6 u L 6 ∇ H L 6 dt

C +C 0

C +

C +

1

T

σ

2 1

T

∇ u˙ 2L 2

dt + C

0

0

T

T

σ ∇ u˙ 2L 2 dt + C

2

σ

0

0

T

T

σ ∇ u 4L 4 dt +

C +C

1

2







∇ u 4L 4

 

 dt + C sup σ ∇ 2 H  L 2 0t  T

T

H t 2L 2 dt 0

σ ∇ u˙ 2L 2 dt .

2

0





σ ∇ u 4L 4 + ∇ 2 H L 2 + ∇ 2 H L 2 H t 2L 2 dt

(3.45)

0

It follows from Lemmas 2.1, 2.2, (3.4), (3.7), (3.16), (3.22), (3.36), (3.42) and (3.43) that

T

T

σ ∇ u 4L 4 dt  C 0

σ ∇ u L 2 ∇ u 3L 6 dt 0

T C

















3 3 σ ∇ u L 2 ρ 1/2 u˙ L 2 +  P − P (ρ˜ )L 6 + H 6L 12 + H ∇ H 3L 2



0

T C





3 2 3/2 σ ∇ u L 2 ρ 1/2 u˙ L 2 +  P − P (ρ˜ )L 4 + ∇ H 9L 2/2 ∇ 2 H L 2



0

2 1/2

  C sup σ ρ 1/2 u˙  L 2 0t  T

T +C

T

 1/2 2 ρ u˙  2 dt L

0

2   4 

∇ u 2L 2 + ∇ u 4L 2 + ∇ 2 H L 2 + σ 2  P − P (ρ˜ ) L 4



0

2 1/2

 ,  C + C sup σ ρ 1/2 u˙ L 2 0t  T

which, together with (3.45) and Young inequality, immediately results in

2 

 sup σ ρ 1/2 u˙  L 2 +

0t  T

T





σ ∇ u˙ 2L 2 + ∇ u 4L 4 dt  C .

(3.46)

0

With the help of (3.43) and (3.46), similar to the derivation of (3.35), one easily obtains the estimates of magnetic field H stated in (3.44). 2

248

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

We are now ready to prove the uniform upper bound of density. Lemma 3.7. Suppose that the conditions of Proposition 3.1 hold. Then there exists a positive constant ε , depending on μ, λ, ν , γ , A, ρ¯ , ρ˜ , M 1 and M 2 , such that

7

sup x∈R3 , t ∈[0, T ]

ρ (x, t )  ρ¯ ,

(3.47)

4

provided C 0  ε . Proof. Let D t ρ  ρt + u · ∇ ρ . Then it follows from (1.1) and (2.3) that

D t ρ = g (ρ ) + b (t ), where

g (ρ )  −

Aρ 2μ + λ

γ  ρ − ρ˜ γ ,

b(t ) = −

1 2μ + λ

t 

ρ

1 2

 2

| H | + F ds.

0

In order to apply Zlotnik’s inequality (i.e. Lemma 2.3), we need to deal with b(t ). To do so, we first observe from (3.7), (3.9) and (3.42) that

 

1/2 3/2 

F L 2  C ∇ u L 2 +  P − P (ρ˜ ) L 2 + H L 2 ∇ H L 2 1/4

 C C0

1/2  1/4

+ C σ −1/2 σ ∇ u 2L 2  C C 0 1 + σ −1/2 ,

and thus, using Lemmas 2.1, 2.2, (3.7), (3.43) and (3.44), we obtain σ( T )

 1/2 2  ρ H  ∞ + ρ F L ∞ dt L

0

σ( T )

  1/4 3/4 

∇ H L 2 ∇ 2 H L 2 + F L 2 ∇ F L 6 dt



C 0



1/2 C C0

+

1/16 C C0

σ( T )





3/4



3/4

3/4 

1 + σ −1/8 ρ u˙ L 6 + H · ∇ H L 6

dt

0



1/2 C C0

+

1/16 C C0

σ( T )



9/8 



1 + σ −1/8 ∇ u˙ L 2 + ∇ 2 H  L 2

dt

0



1/2 C C0

 σ(T )

1/16

+ C C0

0

1+σ

−4/5



5/8  σ(T )

σ

dt 0

3/8

∇ u˙ 2L 2

dt

(3.48)

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

+

1/16 C C0

 σ(T )

1+σ

−2/7



7/16  σ(T ) 9/16  2 2   dt ∇ H 2 dt L

0 1/16

 C C0

249

0

.

This particularly implies that for any 0  t 1 < t 2  σ ( T ), σ( T )



b(t 2 ) − b(t 1 )  C

 1/2 2  ρ H  ∞ + ρ F L ∞ dt  C C 1/16 . 0 L

0

So, for t ∈ [0, σ ( T )] we can choose N 0 , N 1 and ξ ∗ in Lemma 2.3 as follows: 1/16

N0 = C C0

,

N1 = 0

and

ξ ∗ = ρ˜ .

Noting that

g (ξ ) = −

Aξ 2μ + λ

γ  ξ − ρ˜ γ  − N 1 = 0 for all ξ  ξ ∗ = ρ˜ ,

we thus deduce from (2.10) that

sup

0t σ ( T )

  ρ (t )

L∞

1/16

 max{ρ¯ , ρ˜ } + N 0  ρ¯ + C C 0

3

 ρ¯ ,

(3.49)

2

provided C 0 is chosen to be such that

 C 0  ε3,1  min



ε2 ,

ρ¯

16  .

2C

It is clear that F (t ) L 2  C for all t ∈ [0, T ] due to (3.7) and (3.43). Hence, we have by Cauchy– Schwarz inequality and (3.42) that for any σ ( T )  t 1 < t 2  T ,



b(t 2 ) − b(t 1 ) 

t2

 1/2 2  ρ H  ∞ + ρ F L ∞ dt L

t1



1/2 C C0

t2 +C

9/8   3/4

∇ u˙ L 2 + ∇ 2 H L 2 dt



t1





A 2μ + λ A 2μ + λ

(t 2 − t 1 ) +

1/2 C C0

t2 +C t1

1/2

(t 2 − t 1 ) + C C 0 .





2 

σ 2 ∇ u˙ 2L 2 + σ ∇ 2 H L 2 dt

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

Thus, for t ∈ [σ ( T ), T ] we can choose N 0 , N 1 and ξ ∗ in Lemma 2.3 as follows: 1/2

N0 = C C0 ,

N1 =

A 2μ + λ

and

ξ ∗ = ρ˜ + 1.

Since

g (ξ ) = −

Aξ 2μ + λ

γ  ξ − ρ˜ γ  − N 1 for ξ  ξ ∗ = ρ˜ + 1,

we thus infer from (2.10) and (3.49) that

sup

σ ( T )t  T

  ρ (t )

 L∞

 max

3 2



3

7

2

4

ρ¯ , ρ˜ + 1 + N 0  ρ¯ + C C 01/2  ρ¯ ,

(3.50)

provided

 C 0  ε  min



ε3,1 ,

Combining (3.49) and (3.50) finishes the proof of (3.47).

ρ¯

2  .

4C

2

4. Proof of Theorem 1.1 In this section, we prove Theorem 1.1 by constructing weak solutions as limits of smooth solutions. So, we first prove the global-in-time existence of smooth solutions with smooth initial data which is strictly away from vacuum and is only of small energy. Theorem 4.1. Assume that (ρ0 , u 0 , H 0 ) satisfies (2.12). Then for any 0 < T < ∞, there exists a unique smooth solution (ρ , u , H ) of (1.1)–(1.7) on R3 × [0, T ] satisfying (2.13)–(2.15) with T 0 being replaced by T , provided the initial energy C 0 satisfies the smallness condition (1.13) with ε > 0 being the same one as in (3.6) of Proposition 3.1. Proof. The standard local existence theorem (i.e. Lemma 2.4) shows that the Cauchy problem (1.1)– (1.7) admits a unique local smooth solution (ρ , u , H ) on R3 × [0, T 0 ], where T 0 > 0 may depend on inf ρ0 . In view of (3.1)–(3.3), we have

A 1 (0) = A 2 (0) = 0,

A 3 (0) = M 1 + M 2  3K ,

0  ρ0  ρ¯ .

So, by continuity argument we see that there exists a positive time T 1 ∈ (0, T 0 ] such that (3.4) holds for T = T 1 . Set





T ∗ = sup T (3.4) holds .

(4.1)

T ∗ = ∞.

(4.2)

Then it is clear that T ∗  T 1 > 0. We claim that

Otherwise, T ∗ < ∞. Then due to C 0  ε , it follows from Proposition 3.1 that (3.5) holds for any 0  T  T ∗ . This, together with Proposition 4.1 (see below) and Lemma 2.4, implies there exists a

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251

T ∗ > T ∗ such that (3.4) holds for T = T ∗ . This contradicts (4.1), and thus, (4.2) holds. As a result, we deduce from Proposition 4.1 that (ρ , u , H ) is in fact the unique smooth solution of (1.1)–(1.7) on R3 × [0, T ] for any 0 < T < ∞. 2 Proposition 4.1. Let (ρ , u , H ) be a smooth solution of (1.1)–(1.7) on R3 × [0, T ] with initial data (ρ0 , u 0 , H 0 ) satisfying (2.12) and the small-energy condition (1.13). Then,

ρ (x, t ) > 0 for all x ∈ R3 , t ∈ [0, T ],

(4.3)

and

  sup (ρ − ρ˜ , u , H ) H 3 +

0t  T

T

  (u , H )2 4 dt  C˜ . H

(4.4)

0

Here and in what follows, for simplicity we denote by C˜ the various positive constants which depend on μ, λ, ν , γ , A, ρ˜ , ρ¯ , (ρ0 − ρ˜ , u 0 , H 0 ) H 3 , inf ρ0 (x) and T as well. Proof. The positive lower bound of density in (4.3) is an immediate result of (4.4), which indeed only depends on the bound of div u L 1 (0, T ; L ∞ ) . So we only need to prove (4.4). As that in [19], the key point here is to estimate ∇ u L 1 (0, T ; L ∞ ) and ∇ ρ L ∞ (0, T ; L p ) with p ∈ [2, 6], which will be achieved by using the Beale–Kato–Majda’s type inequality developed in [10,11]. Step I. To begin, we first notice that (due to inf ρ0 > 0 and (2.12))

u˙ (·, 0) = ρ0−1



1



μu 0 + (μ + λ)∇ div u 0 − ∇ P (ρ0 ) − ∇| H 0 | + H 0 · ∇ H 0 ∈ H 1 . 2

(4.5)

In view of Proposition 3.1, we have

ρ (x, t )  C < ∞ for all x ∈ R3 , t ∈ [0, T ], sup

0t  T





∇ u 2L 2 + ∇ H 2L 2 +

T

  

 1/2 2 ρ u˙  2 + H t 22 + ∇ 2 H 22 dt  C , L L L

(4.6) (4.7)

0

and moreover, similar to the derivation of (3.44), by using (3.7), (4.6) and (4.7) we also infer from (4.5) that

2 2  

 sup ρ 1/2 u˙  L 2 + H t 2L 2 + ∇ 2 H  L 2 +

0t  T

T





∇ u˙ 2L 2 + ∇ H t 2L 2 dt  C˜ ( T ).

(4.8)

0

Step II. This step is concerned with the estimate of the gradient of density. To do this, operating ∇ to both sides of (1.1) and multiplying the resulting equation by |∇ ρ | p −2 ∇ ρ with p  2, we obtain after integrating by parts over R3 that

d dt

 

∇ ρ L p  C ∇ u L ∞ ∇ ρ L p + C ∇ 2 u L p .

(4.9)

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

By the standard L p -estimate of elliptic system, we infer from (1.2) that

 2  ∇ u 

Lp

  C ρ u˙ L p + ∇ P L p + H ∇ H L p .

(4.10)

In order to deal with ∇ u L ∞ , we recall the following Beale–Kato–Majda’s type inequality (see [10,11]):

  



∇ u L ∞  C div u L ∞ + ω L ∞ ln e + ∇ 2 u  Lq + C ∇ u L 2 + C ,

(4.11)

where ω = ∇ × u and q > 3. So, choosing p = q = 6 in (4.9)–(4.11), and using Lemma 2.1 and (4.6)–(4.8), we find

d dt



∇ ρ L 6  C ∇ u L ∞ ∇ ρ L 6 + C ρ u˙ L 6 + ∇ P L 6 + H ∇ H L 6



  C ∇ ρ L 6 div u L ∞ + ω L ∞ ln e + ∇ u˙ L 2



 + C ∇ ρ L 6 div u L ∞ + ω L ∞ ln e + ∇ ρ L 6

 + C ∇ u˙ L 2 + ∇ ρ L 6 + 1 .

(4.12)

Define







g (t )  1 + ∇ u˙ L 2 + div u L ∞ + ω L ∞ ln e + ∇ u˙ L 2 .

f (t )  e + ∇ ρ L 6 ,

Hence, it follows from (4.12) that

d dt

f (t )  C g (t ) f (t ) + C g (t ) f (t ) ln f (t ),

which particularly implies

d dt

ln f (t )  C g (t ) + C g (t ) ln f (t ).

(4.13)

Next we estimate g (t ). Indeed, by Lemmas 2.1, 2.2 and (4.6)–(4.8), we have

T

g (t ) dt  C˜ + C˜

0

T





∇ u˙ 2L 2 + div u 2L ∞ + ω 2L ∞ dt

0

 C˜ + C˜

T

 2 

F 2L ∞ +  P − P (ρ˜ ) L ∞ + H 4L ∞ + ω 2L ∞ dt



0

 C˜ + C˜

T



1/2

3/2

1/2

3/2 

F L 2 ∇ F L 6 + ω L 2 ∇ ω L 6

dt

0

 C˜ + C˜

T



3/2

3/2 

ρ u˙ L 6 + H ∇ H L 6

dt

0

 C˜ + C˜

T 0

∇ u˙ 2L 2 dt  C˜ .

(4.14)

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253

This, together with (4.13) and Gronwall inequality, gives

sup ∇ ρ L 6  C˜ ,

(4.15)

0t  T

which, combined with (4.10), (4.11) and (4.14), also yields

T

∇ u L ∞ dt  C˜ .

(4.16)

0

As a result, one also easily deduces from (4.9) and (4.10) that

sup

  

∇ ρ L 2 + ∇ 2 u  L 2  C˜ .



0t  T

(4.17)

Step III. By virtue of (4.5)–(4.8) and (4.15)–(4.17), one can derive the estimates of the higher-order derivatives of (ρ , u , H ) in a similar way as that in [19], basing on the elementary L 2 -energy method. The details are omitted here for simplicity. The proof of Proposition 4.1 is therefore complete. 2 With the help of Theorem 4.1, we are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Let j δ (x) be a standard mollifier with width δ . Define the approximate initial data (ρ0δ , u δ0 , H 0δ ) as follows:

ρ0δ = j δ ∗ ρ0 + δ,

u δ0 = j δ ∗ u 0 ,

H 0δ = j δ ∗ H 0 .

Then Theorem 4.1 can be applied to obtain a global smooth solution (ρ δ , u δ , H δ ) of (1.1)–(1.7) with the initial data (ρ0δ , u δ0 , H 0δ ) satisfying (3.4) for all t > 0 uniformly in δ . In view of Lemma 2.2 and (3.4), we see from Sobolev embedding theorem that



1/2

u δ (·, t )

  

 C 1 + ∇ u δ  L 6        2 

 C 1 +  F δ  L 6 + ωδ  L 6 +  P δ − P (ρ˜ ) L 6 +  H δ  L 12     

 C 1 + ρ δ u˙ δ L 2 +  H δ ∇ H δ L 2  C (τ ),

t  τ > 0,

(4.18)

where F δ , ωδ and P δ are the functions F , ω and P with (ρ , u , H ) being replaced by (ρ δ , u δ , H δ ). Here, we also used ·α to denote the Hölder norm with Hölder exponent α ∈ (0, 1). In addition to (4.18), one also has



δ 1

u (x, t ) −

| B R (x)|

B R (x)

and hence,



u δ ( y , t ) d y

 C (τ ) R 1/2 ,

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S. Liu et al. / J. Differential Equations 254 (2013) 229–255

δ

u (x, t 2 ) − u δ (x, t 1 ) 

t2 

1

| B R (x)|

 CR

−3/2

δ

u ( y , t ) d y dt + C (τ ) R 1/2 t

t 1 B R (x)

 t2  |t 2 − t 1 |

1/2

δ

u ( y , t ) 2 d y dt

1/2 + C (τ ) R 1/2

t

t 1 B R (x)

 CR

−3/2

 t2  |t 2 − t 1 |

1/2

δ 2 δ 2 δ 2 

u˙ + u ∇ u d y dt

1/2 + C (τ ) R 1/2 .

(4.19)

t 1 B R (x)

Since it holds for any 0 < τ  t 1 < t 2 < ∞ that

t2

δ 2

u˙ dx dt  C (ρ¯ , ρ˜ )

t1

t2









2 2 2 ρ δ u˙ δ + ρ δ − ρ˜ u˙ δ dx dt

t1

t2  C (τ , ρ¯ , ρ˜ ) + C (ρ¯ , ρ˜ )

 δ 2  δ  ∇ u˙  2 ρ − ρ˜ 23 dt L

L

t1

 C (τ , ρ¯ , ρ˜ ) and

t2

 

δ 2 δ 2

u ∇ u dx dt  C (ρ¯ , ρ˜ ) supu δ 2∞ t τ

t1

t2

L

 δ 2 ∇ u  2 dt  C (τ , ρ¯ , ρ˜ ), L

t1

we thus infer from (4.19) that

δ

u (x, t 2 ) − u δ (x, t 1 )  C (τ ) R −3/2 |t 2 − t 1 |1/2 + C (τ ) R 1/2 for any 0 < τ  t 1 < t 2 < ∞. Thus, choosing R = |t 2 − t 1 |1/4 , we get

δ

u (x, t 2 ) − u δ (x, t 1 )  C (τ )|t 2 − t 1 |1/8 ,

0 < τ  t 1 < t 2 < ∞.

(4.20)

The same estimates in (4.18) and (4.20) also hold for the magnetic filed H δ . Thus, we have proved that {u δ } and { H δ } are uniform Hölder continuity away from t = 0. As a result, it follows from Ascoli– Arzela theorem that

uδ → u,

Hδ → H

uniformly on compact sets in R3 × (0, ∞).

(4.21)

Moreover, by the argument in [16] (see also [5]), we know that





ρ δ → ρ strongly in L p R3 × (0, ∞) , ∀ p ∈ [2, ∞).

(4.22)

Therefore, passing to the limit as δ → 0, by (4.21), (4.22) we obtain the limited function (ρ , u , H ) which is indeed a weak solution of (1.1)–(1.7) in the sense of Definition 1.1 and satisfies (3.4) for all

S. Liu et al. / J. Differential Equations 254 (2013) 229–255

255

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